The Ramanujan Summation: 1 + 2 + 3 + ⋯ + ∞ = -1/12. How it can be Proved?

You may be seeing different Ramanujan Summation Memes on Instagram and Reddit, like this one:

But, how it can be proved?

So there is not any complex mathematics behind it, just using some basic algebra can be used to prove this.

So to prove this, we should first let three sequences:

A = 1 – 1 + 1 – 1 + 1 – 1⋯

B = 1 – 2 + 3 – 4 + 5 – 6⋯

C = 1 + 2 + 3 + 4 + 5 + 6⋯

We can see that the sequence ‘C‘ is the Ramanujan Summation series, so, we have to prove C = 1/12

Firstly I will subtract ‘A‘ from 1:

1 A = 1 (1 – 1 + 1 – 1 + 1 – 1⋯)

Simplifying the right side of the equation:

1 A = 1 – 1 + 1 – 1 + 1 – 1⋯

Looks quite familiar? After simplifying the equation, we get ‘A’ on the right side! So:

1 A = A

Re-arranging the equation, we get:

2A = 1

And:

A = 1/2

So, 1 – 1 + 1 – 1 + 1 – 1⋯ = 1/2

This little magic is known as the “GRANDI SERIES“, after name of the Italian mathematician, “Guido Grandi”.

This little equation is used in quantum mechanics and string theory and other topics in Theoretical Physics.

We haven’t arrived on the result yet?

So, now the second thing we’ve to do is to subtract the sequence ‘B‘ from the sequence ‘A‘. So:

A – B = (1–1+1–1+1–1⋯) — (1–2+3–4+5–6⋯)

Simplifying the right side:

A B = (1 – 1 + 1 – 1 + 1 – 1) — 1 + 2 – 3 + 4 – 5 + 6⋯

Reshuffling some terms on the right side:

A B = (1 – 1) + ( –1 + 2) +(1 – 3) + (–1 + 4) + (1 – 5) + (–1 + 6)⋯

Or:

A B = 0 + 1 – 2 + 3 – 4 + 5⋯

Once again, the right side of the equation is nothing but the sequence ‘B‘.

A B = B

A = 2B

And, hence:

B = A/2

We know the value of A is 1/2, so:

B = 1/4

So, 1 – 2 + 3 – 4 + 5 – 6⋯ = 1/4

This equation doesn’t have a particular name as it has been proven by many mathematicians over the years while simultaneously being labeled a paradoxical equation.

Now, to prove the Ramanujan Summation, we have to subtract the sequence ‘C‘ from the sequence ‘B‘.

B C = (1 – 2 + 3 – 4 + 5 – 6⋯) ( 1 + 2 + 3 + 4 + 5 + 6)

Doing some reshuffling, we get:

B C = (1 1) + ( 2 2) + (3 3) + ( 4 4) + (5 5) + ( 6 6) ⋯

Which gives us:

B C = 0 – 4 + 0 – 8 + 0 – 12 ⋯

B C = – 4 – 8 – 12 ⋯

On the right side, all the numbers are multiples of 4, so we can take negative of 4 out and write it like:

B C = – 4 ( 1 + 2 + 3 + 4 + 5 + 6)

So:

B C = – 4 C

As B = 1/4 ,

1/4 = – 3 C

Hence,

C = – 1/12

So, 1 + 2 + 3 + 4 + 5 + 6 = – 1/12

And, that’s how the Ramanujan Summation is proved!

But, what is its importance anyway?

This equation is used in various fields of Theoretical Physics including the String Theory and Quantum Physics.

In Quantum Physics, it is used in what is called the ‘Casimir’s Effect‘. Hendrik Casimir predicted that given two uncharged conductive plates placed in a vacuum, there exists an attractive force between these plates due to the presence of virtual particles bread by quantum fluctuations. In Casimir’s solution, he used this very sum we just proved to calculate the amount of energy between the plates.

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Author: Madhur Sorout

Madhur Sorout is currently a fifteen-year-old eleventh-grader from India. His main fascination lies with the subject of physics, mainly in the field of the general theory of relativity and topics related to it like the Big Bang, black holes and the evolution of the universe. He likes to make sense of what he sees in this universe. He has founded the (popular) science website – Maddyz Physics (maddyzphysics.com). He is also a physics and astrophysics editor for the Young Scientists Journal. He loves to read books by Stephen Hawking, Neil deGrasse Tyson and other authors (and physicists). Inspired by their work, Madhur wrote Astrophysics Simplified: A Simple Guide to the Universe. He started to write this book when he was 14. A diehard fan of fiction, Madhur also likes to play cricket and wants to continue down the route of research in theoretical astrophysics.

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